If ( a k ) k = 1 ∞ converges to a ∈ R , then lim n → ∞ 1 n ∑ k = 1 n a k = a

Question

If   (      a    k        )          k      =      1              ∞       converges to   a  ∈      R  , then       lim          n      →      ∞            1    n        ∑          k      =      1              n            a    k    =  a

Maggie Bowman · Accepted Answer

This is proven quite straitforward: Let   ϵ  &amp;gt;  0. Scince       a    k    →  a, we can choose   m  ∈      N  , such that       |        a    n    −  a      |    ≤      ϵ    2   for all   n  ≥  m. Then we can choose   N  ∈      N  , such that      1    N        ∑          j      =      1        m        |        a    j        |    ≤      ϵ    2    .Then, for   n  ≥  N      |          1      n              (              ∑                  j          =          1                n                    a        j            )        −    a    |    ≤      1    n        ∑          j      =      1        m        |        a    j        |    +      1    n        ∑          j      =      m        n        |        a    j    −  a      |    ≤      ϵ    2    +            n      −      m      +      1        n        ϵ    2    ≤  ϵ  .However, your intuition is good, keep it in mind. The proof basically is just your intuition in formulas. (Your       a    k    ≈  a becomes       |        a    k    −  a      |    ≤      ϵ    2  )

gorgeousgen9487 · Accepted Answer

If you take       b    n    =  n, and       s    n   as the partial sums       ∑          k      =      1        n        a    k   of your original sequence, then the conditions of the Stolz-Cesaro theorem are satisfied. Then consider the limit   (  1  )       b          n      +      1        −      b    n    =  1  ,  ∀  n, so we get:      lim          n      →      ∞            s          n      +      1        −      s    n    =      lim          n      →      ∞            a          n      +      1        =  aAnd we are done:      lim          n      →      ∞            a          n      +      1        =  a  =      lim          n      →      ∞            1    n        ∑          k      =      1        n        a    k

If ( a k </msub> <msubsup> ) <mrow class="MJX-TeXAtom-ORD"> k

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